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Question

If BC is a diameter of a circle with centre O and OD is perpendicular to the chord AB of a circle, show that CA=2 OD.   


Solution


Given : A circle with centre O, diameter BC and OD perpendicular to the chord AB.

To prove : CA=2 OD

Proof: Since OD is to chord AB
As we know that the perpendicular drawn from the centre to a chord bisects the chord.
So, the point D is bisect the chord AB
Thus, D is the midpoint of AB.
and also O act as the centre and the mid point of diameter BC.

In ΔABC,O and D are the mid points of BC and AB respectively
ODAC and OD=CA2
(By midpoint theorem, segment joining the mid points of two sides of a triangle is half of the third side)
CA=2OD
Hence, proved.

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